3.22 \(\int \frac{(A+B x) (b x+c x^2)^2}{x^4} \, dx\)

Optimal. Leaf size=44 \[ -\frac{A b^2}{x}+c x (A c+2 b B)+b \log (x) (2 A c+b B)+\frac{1}{2} B c^2 x^2 \]

[Out]

-((A*b^2)/x) + c*(2*b*B + A*c)*x + (B*c^2*x^2)/2 + b*(b*B + 2*A*c)*Log[x]

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Rubi [A]  time = 0.0315554, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {765} \[ -\frac{A b^2}{x}+c x (A c+2 b B)+b \log (x) (2 A c+b B)+\frac{1}{2} B c^2 x^2 \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2)^2)/x^4,x]

[Out]

-((A*b^2)/x) + c*(2*b*B + A*c)*x + (B*c^2*x^2)/2 + b*(b*B + 2*A*c)*Log[x]

Rule 765

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand
Integrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (
GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int \frac{(A+B x) \left (b x+c x^2\right )^2}{x^4} \, dx &=\int \left (c (2 b B+A c)+\frac{A b^2}{x^2}+\frac{b (b B+2 A c)}{x}+B c^2 x\right ) \, dx\\ &=-\frac{A b^2}{x}+c (2 b B+A c) x+\frac{1}{2} B c^2 x^2+b (b B+2 A c) \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0272924, size = 43, normalized size = 0.98 \[ A \left (c^2 x-\frac{b^2}{x}\right )+b \log (x) (2 A c+b B)+\frac{1}{2} B c x (4 b+c x) \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2)^2)/x^4,x]

[Out]

(B*c*x*(4*b + c*x))/2 + A*(-(b^2/x) + c^2*x) + b*(b*B + 2*A*c)*Log[x]

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Maple [A]  time = 0.005, size = 46, normalized size = 1.1 \begin{align*}{\frac{B{c}^{2}{x}^{2}}{2}}+A{c}^{2}x+2\,Bbcx+2\,A\ln \left ( x \right ) bc+{b}^{2}B\ln \left ( x \right ) -{\frac{A{b}^{2}}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)^2/x^4,x)

[Out]

1/2*B*c^2*x^2+A*c^2*x+2*B*b*c*x+2*A*ln(x)*b*c+b^2*B*ln(x)-A*b^2/x

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Maxima [A]  time = 2.6958, size = 62, normalized size = 1.41 \begin{align*} \frac{1}{2} \, B c^{2} x^{2} - \frac{A b^{2}}{x} +{\left (2 \, B b c + A c^{2}\right )} x +{\left (B b^{2} + 2 \, A b c\right )} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/x^4,x, algorithm="maxima")

[Out]

1/2*B*c^2*x^2 - A*b^2/x + (2*B*b*c + A*c^2)*x + (B*b^2 + 2*A*b*c)*log(x)

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Fricas [A]  time = 1.7248, size = 116, normalized size = 2.64 \begin{align*} \frac{B c^{2} x^{3} - 2 \, A b^{2} + 2 \,{\left (2 \, B b c + A c^{2}\right )} x^{2} + 2 \,{\left (B b^{2} + 2 \, A b c\right )} x \log \left (x\right )}{2 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/x^4,x, algorithm="fricas")

[Out]

1/2*(B*c^2*x^3 - 2*A*b^2 + 2*(2*B*b*c + A*c^2)*x^2 + 2*(B*b^2 + 2*A*b*c)*x*log(x))/x

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Sympy [A]  time = 0.354409, size = 42, normalized size = 0.95 \begin{align*} - \frac{A b^{2}}{x} + \frac{B c^{2} x^{2}}{2} + b \left (2 A c + B b\right ) \log{\left (x \right )} + x \left (A c^{2} + 2 B b c\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)**2/x**4,x)

[Out]

-A*b**2/x + B*c**2*x**2/2 + b*(2*A*c + B*b)*log(x) + x*(A*c**2 + 2*B*b*c)

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Giac [A]  time = 1.12026, size = 62, normalized size = 1.41 \begin{align*} \frac{1}{2} \, B c^{2} x^{2} + 2 \, B b c x + A c^{2} x - \frac{A b^{2}}{x} +{\left (B b^{2} + 2 \, A b c\right )} \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/x^4,x, algorithm="giac")

[Out]

1/2*B*c^2*x^2 + 2*B*b*c*x + A*c^2*x - A*b^2/x + (B*b^2 + 2*A*b*c)*log(abs(x))